Reid opted for the so-called "nuclear option" to help win confirmation of the three court nominees. The option was a series of rules changes that allowed confirmation with a simple majority instead of the required 60 senators. The Nevada senator's action removed the ability of senators to filibuster a non-Supreme Court judicial nominee.

Reid went to that length after Republicans resisted the Obama administration's move to fill the three vacant seats, with some GOP members arguing the court's workload was so light it didn't need the three judges.

Those new judges—Patricia Millett, Robert Wilkins and Nina Pillard—tipped the balance of power on the U.S. Court of Appeals for the District of Columbia Circuit in favor of Democratic appointees by a 7-4 margin. They joined a fourth Obama appointee, Sri Srinivasan, whose nomination had sailed through in May 2013.

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Sen. Harry Reid

Their numerical advantage over their Republican-appointed colleagues could soon prove decisive on a huge threat to Obamacare and its goal of offering affordable insurance to millions of uninsured people.

The full circuit will be asked by the Obama administration to go over Tuesday's ruling that found billions of dollars worth of subsidies given to nearly 5 million people who enrolled in insurance plans via the HealthCare.gov federal exchange were illegal. Such a reconsideration is called an "en banc" review.

The main reason Obamacare advocates are talking confidently of their chances of getting the full circuit to reverse that ruling, which affects people in 36 states, is their belief that the majority of Democratic nominees will vote that way.

"I would agree with that," said Timothy Jost, a Washington and Lee University School of Law professor who has been a leading Obamacare advocate.

Another law professor, who is one of the intellectual architects of the legal challenge to the Obamacare subsidies, also said he expects his side might ultimately lose at the D.C. circuit level because of the political makeup of the bench.

"I think having more liberal judges on the court—that is, more judges who might be inclined to adopt a broad, purposive reading of the statute even where this creates problems for the express statutory text—helps the administration," Case Western Reserve University professor Jonathan Adler said.

"That such judges, being new to the court, are also less familiar with the D.C.Circuit's traditional, narrow view of when 'en banc' review is appropriate also helps the administration."

In its ruling Tuesday, the three-judge panel of the D.C. circuit ruled that taxpayer-funded credits received by 4.7 million HealthCare.gov enrollees to help pay for their Obamacare plans were illegal because they were granted through the federal exchange, as opposed to one set up by a state. The decision is based on explicit language in the Affordable Care Act that authorizes the subsidies to be granted through an exchange established by a state—the statue is mum on the issuance of subsidies issued through a federally run exchange.

Bill Clark | CQ Roll Call

File photo: Patricia Ann Millett, nominee to be U.S. circuit judge for the District of Columbia Circuit, speaks with chairman Patrick Leahy, D-Vt., before the Senate Judiciary Committee hearing on the nominations of four judges on Wednesday, July 10, 2013.

If upheld as law, that decision would not only revoke billions of dollars worth of subsidies. It also would effectively destroy the Obamacare rule in 36 states that people obtain health insurance or pay a fine, as well as the looming rule that will require many employers to offer affordable coverage to workers or pay a fine themselves.

The Obama administration immediately announced Tuesday that it will seek to have the decision overturned by the entire D.C. appeals circuit.

The administration argues that the panel's decision was wrong because it ignores the overall intent of the ACA, to make insurance affordable for uninsured people throughout the U.S.

The en banc review panel would include not only the 11 active judges on the circuit, but also the two senior judges who were on the panel that issued Tuesday's decision.

One of the senior judges, Raymond Randolph, was part of the majority opinion written by Judge Thomas Griffith that found the subsidies were illegal. The other senior judge on the panel, Harry Edwards, strongly dissented.

Randolph was appointed by a President George H.W. Bush in 1990, while Edwards was appointed by President Jimmy Carter in 1980.

If they each voted the same way during an en banc review, and if the rest of the panel broke according to the parties of the presidents that appointed them, then the vote would be 8 to 5 to overturn the panel's decision.

Without the three judges that Reid pushed through, the decision would likely have been 5-5, and "a tie would have upheld the ruling," Jost said.

The review not only gives the administration a chance to reverse Tuesday's ruling, it also gives it a chance to reduce the likelihood that the issue will ever be decided by the Supreme Court. That's because the Supreme Court is considered much less likely to take such a case if there is no split in opinion between federal appeals circuits, which review decisions made federal district trial courts in their geographical regions.